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The dynamics of the jellyfish joyride: mathematical discussion of the causes leading to blooming. (English) Zbl 1341.37059
Summary: Dramatic increases in jellyfish populations, which lead to the collapse of formally healthy ecosystems, are repeatedly reported from many different sites. Because of their devastating effects on fish populations, understanding of the causes for such bloomings are of major ecological as well as economical importance. Based on a previous work, we set up a two-dimensional predator-prey model for the fish-jellyfish interactions by taking fishery as well as environmental conditions into account. It assumes the fish as the dominant predator species. By totally analytic means, we completely classify all equilibria in terms of existence and non-linear stability and give a description of this system’s non-linear global dynamics. We analytically study the non-equilibrium dynamics and detect homoclinic, Andronov-Hopf, and Takens-Bogdanov bifurcations as well as the non-existence of periodic and homoclinic orbits in certain relevant regions of the phase space. A hereby found, and analytically rigorously stated, funnel phenomenon gives rise to a mathematical explanation of jellyfish blooming beyond the typical one in terms of large-amplitude limit cycles based on the well-known Rosenzweig-MacArthur equations. Also, we illustrate the plethora of bifurcation scenarios by numerical results.
##### MSC:
 37N25 Dynamical systems in biology 37G10 Bifurcations of singular points in dynamical systems 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 37C75 Stability theory for smooth dynamical systems 34C23 Bifurcation theory for ordinary differential equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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